## Overview

This school will present a range of lectures on quantum structures in physics and computer science, with a focus on abstract algebraic techniques, including category theory. It is an ideal event for PhD students, as well as more established researchers, who would like to learn more about these exciting topics close to the research frontier.The courses will be accessible to anyone who has taken a first course in quantum information. Everyone is welcome to participate. Registration information can be found below.

This event is twinned with a conference in honour of Prakash Panangaden on the occasion of his 60th birthday, to be held on the dates 23--25 May 2014 at the University of Oxford: http://www.cs.ox.ac.uk/pf2014

Please print out the event poster for display in your own institution.

The school will feature the following courses:

- Samson Abramsky. Contextual semantics: quantum mechanics and beyond
- Jon Barrett. Correlations and contexts: the quantum no-go theorems
- Bart Jacobs. Kadison duality
- Prakash Panangaden. Stone, Gelfand and Pontryagin dualities
- Peter Selinger. Number-theoretic methods in quantum information theory
- Bob Coecke and Aleks Kissinger. Picturing quantum processes
- Chris Heunen and Jamie Vicary. Categorical quantum mechanics

Samson Abramsky. Contextual semantics: quantum mechanics and beyond

We shall show how quantum non-locality and contextuality are naturally described and unified in the language of sheaf theory. This leads to several concrete developments in quantum information and foundations:

- a novel classification of multipartite entangled states in terms of

their degree of non-locality - a topological analysis of entanglement monogamy and macroscopic locality
- a cohomological characterisation of contextuality
- a unifying principle for Bell inequalities based on logical

consistency conditions

- relational database theory
- constraint satisfaction
- natural language semantics

Abstract to be announced.

Bart Jacobs. Kadison duality

There is a little known duality in functional analysis that is appropriately called Kadison duality. It connects complete order unit spaces and convex compact Hausdorff spaces. This connection gives a precise formalisation of the duality in terms of C*-algebras. The lectures present the essential steps of the proof of Kadison duality---which are scattered around in the literature---and also show where the duality is relevant. Familiarity is assumed with basic category theory, linear algebra, and topology. C*-algebras will be used in the examples.

Prakash Panangaden. Stone, Gelfand and Pontryagin dualities

The word “duality” is tossed around in almost all areas of mathematics and physics (and even parts of computer science). Roughly speaking it allows one to view two, apparently different, mathematical universes as “mirror images of each other.” In mathematics the most venerable and well-known such duality is the familiar duality between a vector space and the space of linear functionals on a vector space. More sophisticated versions of this arise in functional analysis. In physics one talks about dualities between electric and magnetic fields and more sophisticated versions of that duality arise in non-abelian gauge theories.

In this series of three lectures I will discuss the three dualities mentioned in the title. Stone duality relates Boolean algebras with certain kinds of topological spaces and is fundamental for logic. Variations of this duality arise throughout theoretical computer science. The second one on the list is more functional-analytic in nature and relates compact Hausdorff spaces to commutative unital C*-algebras. This is a stepping stone towards the much deeper dualities of interest in quantum mechanics. The third duality, Pontryagin duality, is what underlies Fourier theory and will involve group theory.

A little bit of algebra (groups, rings, algebras) and topology (compactness, separation axioms) will be useful background. The categorical version of the dualities will be given but the category theory used will be elementary, say up to the level of knowing what “adjoint functor” means.

Peter Selinger. Number-theoretic methods in quantum information theory

In these lectures, I will discuss the new class of number-theoretic algorithms in quantum information theory that have come out in the last two years or so. These algorithms solve the well-known problem of decomposing a given unitary operator into gates of a specified gate set, either exactly (known as exact synthesis), or within some given accuracy (known as approximate synthesis). For almost 20 years, the standard solution to this problem had been the Solovay-Kitaev algorithm, which is based on geometric methods of approximation by repeated refinement. For example, in the case of approximate synthesis of single-qubit operators, the Solovay-Kitaev algorithm achieves gate counts of O(log^c(1/epilson)), where c is a constant greater than 3. By contrast, the new number-theoretic algorithms achieve gate counts of O(log(1/epsilon)), and in some cases these algorithms can even be shown to be optimal in an absolute sense. I will introduce the relevant concepts from algebraic number theory, review the current state of the art in both exact and approximate synthesis for the Clifford+T gate base, and comment on open problems in this still very young field.

Bob Coecke and Aleks Kissinger. Picturing quantum processes

We present quantum theory, including quantum computing and quantum foundations, within an entirely diagrammatical formalism. In particular, we give a detailed presentation of the diagrammatic language involved, as well as of how to think of quantum theory as a theory of processes. The material follows a forthcoming book with the same title as this course, and avoids any reference to category theory (but the connection will be explained by other lectures at the school).

Chris Heunen and Jamie Vicary. Categorical quantum mechanics

These lectures cover monoidal categories and their use in modelling the flow of classical and quantum information in quantum mechanics. The lectures will cover monoidal categories and the graphical calculus; Frobenius algebras and their relation to operator algebras; quantum and classical information channels as completely positive maps between algebras; and 2-categories and their applications to quantum computation.

## Registration & Costs

Please contact Destiny Chen if you wish to participate. The deadline for funding applications has passed, but registration is still possible.Conference fee is £50 per person for speakers and participants. Cash (in sterling pounds) payable at the arrival registration on 19th May 2014.

Limited funding is available for students and young researchers, on travel, accommodation and subsistance, out of EPSRC (Grant Reg: EP/I03596X/1). Please contact Destiny Chen for further details prior to your booking.